[EM] Free riding

[Raph Frank]

Ok, I think I have an example which shows the vote management resistance.

Assume the following

2 Seats to be filled and 3 candidates

A1: Popular candidate (party A)
A2: Other candidate (party A)
B: Party B candidate

Honest rankings are

A1's personal supporters

12: A1>B>A2
26: A1>A2>B

Party A's supporters
12: A1>A2>B
13: A2>A1>B

Party B's supporters
27: B

Standard PR-STV

Quota = 30

Round 1
A1: 50
A2: 13
B: 27

A1 elected with 20 surplus (15 for A2 and 5 for B)

Round 2
A1: 30(-20) Elected
A2: 28(+15)
B: 32(+5)

B elected

Result: (A1,B) wins

Assuming vote management.
Party A tells all supporters to vote A2>A1>B

Round 1
A1: 38
A2: 25
B: 27

A1 elected with 8 surplus (2.5 for B and 5.5 for A2)

Round 2
A1: 30(-8)
A2: 30.5(+5.5)
B: 29.5(+2.5)

A2 is elected

Result: (A1,A2)

Vote management has paid off

Schulze's method

A1's personal supporters

12: A1>B>A2
26: A1>A2>B

Party A's supporters
12: A1>A2>B
13: A2>A1>B

Party B's supporters
27: B

Compare (A1,A2) against (A1,B*)

B is the test candidate.

12 prefer A1 to B (Assign to A1's group)
0 prefer A2 to B
51 prefer both to B (Assign 19.5 to A1 and 31.5 to A2)

All groups are the same size of 31.5

Compare (A1,B) against (A1,A2*)

A2 is the test candidate

38 prefer A1 to A2 (assign to A1)
27 prefer B to A2 (assign to B)
12 prefer both to A2 (assign 0.5 to A1 and 11.5 to B)

All groups are the same size of 38.5

This means that (A1,B) beats (A1,A2) by 37.5 votes to 32.5 votes.

Schulze's method under vote management

A1's personal supporters

12: A1>B>A2
26: A1>A2>B

Party A's supporters
25: A2>A1>B

Party B's supporters
27: B

Compare (A1,A2) against (A1,B*)

B is the test candidate.

12 prefer A1 to B (Assign to A1's group)
0 prefer A2 to B
51 prefer both to B (Assign 19.5 to A1 and 31.5 to A2)

All groups are the same size of 31.5

Compare (A1,B) against (A1,A2*)

A2 is the test candidate

26 prefer A1 to A2 (assign to A1)
27 prefer B to A2 (assign to B)
12 prefer both to A2 (assign 6.5 to A1 and 5.5 to B)

All groups are the same size of 32.5

This means that (A1,B) beats (A1,A2) by 37.5 votes to 32.5 votes.

Thus, even under vote management A1,B wins.

The condition seems to be that vote management will still payoff under
Schulze if

A1's personal vote + Party A's vote > A1's personal vote + Party B's vote

thus

if Party A's vote > Party B's vote, then party A can get the
additional seat, but only if vote management is used.